Gaussian mixture models (GMM) can be seen as the probabilistic counterparts of the k-means clustering algorithm. Weighted k-means takes a set of weighted samples and arranges the centroids according to weighted means of the data clusters, where the weights are the weights of the samples. I wonder if there is a GMM-like probabilistic counterpart of weighted k-means.

3 Answers
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pKNN+AL (Jain and Kapoor, 2009) is a probabilistic modification of the KNN classifier. Given a set of points $\{x_1, \ldots, x_n\}$ from $\mathbb{R}^d$, labels $\{y_1, \ldots, y_n\}$ from $[1,C]$, and a Mercer kernel $K$, the probability of $x$ belonging to class $c$ is

I think our notion of the term weight is different. The weight $w_i$ of sample $x_i$ in the sense of weighted k-means is a measure for the importance of $x_i$ for the entire clustering. The weights are fixed and given with the data set a priori. Thus the k-means centriods are centered above clusters the samples of which have high weights. If there are clusters with low weights, they are ignored.
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chrivoMar 26 '10 at 7:56